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Metamath Proof Explorer

Theorem usgr1v

Description: A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017) (Revised by AV, 18-Oct-2020)

Ref Expression
Assertion usgr1v ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )

Proof

Step Hyp Ref Expression
1 usgr1vr ⊢ ( ( 𝐴 ∈ V ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph → ( iEdg ‘ 𝐺 ) = ∅ ) )
2 1 adantrl ⊢ ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( 𝐺 ∈ USGraph → ( iEdg ‘ 𝐺 ) = ∅ ) )
3 simplrl ⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ 𝑊 )
4 simpr ⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → ( iEdg ‘ 𝐺 ) = ∅ )
5 3 4 usgr0e ⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ USGraph )
6 5 ex ⊢ ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( ( iEdg ‘ 𝐺 ) = ∅ → 𝐺 ∈ USGraph ) )
7 2 6 impbid ⊢ ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
8 7 ex ⊢ ( 𝐴 ∈ V → ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) ) )
9 snprc ⊢ ( ¬ 𝐴 ∈ V ↔ { 𝐴 } = ∅ )
10 simpl ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → 𝐺 ∈ 𝑊 )
11 simprr ⊢ ( ( { 𝐴 } = ∅ ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( Vtx ‘ 𝐺 ) = { 𝐴 } )
12 simpl ⊢ ( ( { 𝐴 } = ∅ ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → { 𝐴 } = ∅ )
13 11 12 eqtrd ⊢ ( ( { 𝐴 } = ∅ ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( Vtx ‘ 𝐺 ) = ∅ )
14 usgr0vb ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
15 10 13 14 syl2an2 ⊢ ( ( { 𝐴 } = ∅ ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )
16 15 ex ⊢ ( { 𝐴 } = ∅ → ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) ) )
17 9 16 sylbi ⊢ ( ¬ 𝐴 ∈ V → ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) ) )
18 8 17 pm2.61i ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) )